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Cyclic decompostions

by Bhatia
Tags: linear alegbra, operators
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Bhatia
#1
Feb27-12, 08:53 AM
P: 11
Let T be a linear operator on the the finite dimensional space V, and let R be the range of T.

(a) Prove that R has a complementary T-invariant subspace iff R is independent of the null space N of T.

(b) If R and N are independent, prove that, N is the unique T-invariant subspace complementary to R.

I supposed R has a complementary T - invariant subspace, say, W. Then , R should be T- admissible. I assumed to the contrary, that R intersection T is not equal to {0}. I took a point in the intersection but could not proceed further. Please suggest.
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arkajad
#2
Feb27-12, 11:15 AM
P: 1,411
Can you define "T-admisible"?
Bhatia
#3
Feb27-12, 11:43 AM
P: 11
Quote Quote by arkajad View Post
Can you define "T-admisible"?
Thanks for your reply.

Given a linear operator T on a vector space V then a subspace W is T- admissible if

i) W is invariant under T
(ii) if f(T) β belongs to W, there exists a vector η in W such that f(T)β = f(T) η.

arkajad
#4
Feb27-12, 12:55 PM
P: 1,411
Cyclic decompostions

What is f(T) in (ii)?
morphism
#5
Feb28-12, 04:36 PM
Sci Advisor
HW Helper
P: 2,020
Hint: Fix a basis for the range of T and extend it to a basis for V. What can you say about these extra basis vectors?


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